System and Method For Multiple Frozen-Parameter Dynamic Modeling and Forecasting

ABSTRACT

A system and method is disclosed for determining multiple frozen-parameter dynamic modeling and forecasting of future data values from data values in a data set. Model parameter values are dynamically updated utilizing a time-varying system property, and an updated model is optimally evolved that takes into account the structural changes that may have influenced the actual process, thereby yielding a superior modeling capability. The resultant model is updated in a closed-loop manner. In one exemplary embodiment, the data set comprises financial portfolio data. In another exemplary embodiment, the data set comprises seismic data.

CROSS-REFERENCE TO RELATED PATENT APPLICATION

The present patent application is related to U.S. Provisional PatentApplication Ser. No. 61/448,598, filed Mar. 2, 2011, entitled “MultipleFrozen-Parameter Dynamic Modeling And Forecasting Algorithm,” inventedby N. N. Puri, the disclosure of which being incorporated by referenceherein.

BACKGROUND

Mathematical methods developed in last 20 years have played an importantrole in the study of economics and financial markets. The mathematicalmethods are possible because of cheap and ample computing resourcesbeing available at will. Nevertheless, even the most powerfulcomputational facilities have limitations if the amount of the data isastronomical and the system dynamics are changing continuously. Manyideas relating to dynamic modeling and forecasting algorithms weredeveloped during the hay days of Aerospace advances. In the existingmethods in the literature, the portfolio model parameters are determinedfrom the incoming data and then passively used to predict the futureportfolio values.

For example, Box and Jenkins popularized a three-stage method aimed atselecting an appropriate ARIMA(p,q), model for the purpose of estimatingand forecasting a time series in which p is the order of the system andq represents the number of error terms. This was characterized as modelidentification, i.e., deciding on the order of p,q, estimation involvingthe parameter fitting in the ARIMA model. The time-series modelingcapabilities for linear regression models by ARIMA have been extensivelyused by the Census Bureau. Nevertheless, the methodology of Box andJenkins does not provide a dynamically updated scheme.

BRIEF DESCRIPTION OF THE DRAWINGS

The subject matter disclosed herein is illustrated by way of example andnot by limitation in the accompanying figures in which like referencenumerals indicate similar elements and in which:

FIG. 1 depicts one exemplary embodiment of a flow diagram of the DynamicModeling and Forecasting Algorithm (DMFA) according to the subjectmatter disclosed herein;

FIGS. 2-5 depict graphical comparisons between actual Standard & Poor's500 values and predicted values for different m and n;

FIG. 6 depicts an exemplary embodiment of an article of manufacturecomprising a non-transitory computer-readable medium having storedthereon instructions that, if executed, result in at least the subjectmatter disclosed herein; and

FIG. 7 depicts a functional block diagram of one exemplary embodiment ofan information-handling system capable of providing multiplefrozen-parameter dynamic modeling and forecasting according to thesubject matter disclosed herein.

DETAILED DESCRIPTION

As used herein, the word “exemplary” means “serving as an example,instance, or illustration.” Any embodiment described herein as“exemplary” is not to be construed as necessarily preferred oradvantageous over other embodiments. Additionally, it will beappreciated that for simplicity and/or clarity of illustration, elementsillustrated in the figures have not necessarily been drawn to scale. Forexample, the dimensions of some of the elements may be exaggeratedrelative to other elements for illustrative clarity. Further, in somefigures only one or two of a plurality of similar elements indicated byreference characters for illustrative clarity of the figure, whereas allof the similar element may not be indicated by reference characters.Further still, it should be understood that although some portions ofcomponents and/or elements of the subject matter disclosed herein havebeen omitted from the figures for illustrative clarity, goodengineering, construction and assembly practices, are intended.

The subject matter disclosed herein relates to a system and method fordetermining multiple frozen-parameter dynamic modeling and forecasting,which is a technique referred to herein as the Dynamic Modeling andForecasting Algorithm (DMFA). More specifically, the subject matterdisclosed herein provides an advantage of dynamically updating the modelparameter values utilizing a time-varying system property and evolvingan optimally updated model that takes into account the structuralchanges that may have influenced the actual process, thereby yielding asuperior modeling capability. The resultant model is updated in aclosed-loop manner.

The technique of the subject matter disclosed herein, unlike ARIMA, isdynamic and has many applications across diverse fields, such asaerospace, seismography, and stock-bond portfolio valuation andoptimization. For example, in aerospace engineering, the predicted modelcan be used to make corrections of control system parameters. As anotherexample in seismography, earth-quake predictions are an immensely usefultool, and the model determined by the subject matter disclosed hereincan be used as a powerful predictor of an impending disaster. For astock and bond portfolio, the projections of variable vectors (such asinterest rates, home price index appreciation, forex-rates, swaps) canbe modeled by the subject matter disclosed herein for portfoliovaluation and optimization and used for buying/selling individualentities for a well-balanced portfolio. In a nut-shell, the subjectmatter disclosed herein relates to a technique for dynamically computingthe “frozen” parameters from existing data and predicting the next datapoint. A recursive matrix inversion algorithm results in an efficientcomputation. Moreover, the subject matter disclosed herein can beextended to provide an optimal allocation of capital resources. One ofthe major drawbacks that comes with trading complicated financialinstruments, such as options and financial derivatives, is extremevolatility. The subject matter disclosed herein can be used todynamically predict the volatility parameter, which can be used inBlack-Scholes equations for option trading. Additionally, the subjectmatter disclosed herein provides a methodology to compute optimal optionpricing and the most profitable mortgages for a hedge fund portfolio.

FIG. 1 depicts one exemplary embodiment of the Dynamic Modeling andForecasting Algorithm (DMFA) according to the subject matter disclosedherein. Given a data set {d(i)}_(i=0) ^(n), an (m+1) parameter filtercan be designed such that (n−k)≧(2m+1), in which k represents a minimumnumber of initial samples required before prediction can begin. The(m+1) filter parameters depend upon the instant n and are considered“frozen” in the sense that they are considered as constant from instantk to n, and thereafter time varying, dependent on n, and dynamicallycomputed and updated. The starting instant k can be varied dependingupon the conditions of the modeled process.

Filter Dynamic Model

The modeling process provided by the subject matter disclosed herein isdefined as:

d(n+i−l)=a ₀(n)+d(n−i−l)a _(l)(n)+ . . . +d(n+1−i−l m)a _(m)(n)+ε_(l)(n)for i=0, 1, 2, . . . , n−k; l=1, 2, . . . , m+1; n≧k+2m+1, k≧2m+1.   (1)

The coefficients {a_(i)(n)}_(l=1) ^(m+1) represent the filter modelparameters. The variables {ε_(i)(n)}_(i=k) ^(n) represent random noisehaving zero mean and being uncorrelated with the data (a practicalassumption).

Let

d(n+1−i−l)−d(n+1−i−l−1)=x(n+1−i−l)ε_(i)(n+1)−ε_(i)(n)=η_(i)(n).   (2)

Equation 2 yields m equations involving m parameters {a_(i)(n)}_(l=1)^(m+1). The parameter a₀(m) is eliminated, but will be recovered later.From Eq. 1 and 2, the resulting equations are:

$\begin{matrix}{{{{x\left( {n + 1 - i - l} \right)} = {{\sum\limits_{j = 1}^{m}{{x\left( {n + 1 - i - l - j} \right)}{a_{j}(n)}}} + {\eta_{i}(n)}}},{l = 1},2,\ldots \mspace{14mu},{m;}}{{i = k},{k + 1},\ldots \mspace{14mu},{n.}}} & (3)\end{matrix}$

Multiplying Eq. 3 by x(n−i) and summing from i=0 to i=n−k, we obtain

$\begin{matrix}{{\sum\limits_{i = 0}^{n - k}\left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l} \right)}} \right\rbrack} = {{\sum\limits_{j = 1}^{m}{\sum\limits_{i = 0}^{n - k}{\left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - {l\; j}} \right)}} \right\rbrack {a_{j}(n)}}}} + {\sum\limits_{i = 0}^{n = k}{\left\lbrack {{x\left( {n - i} \right)}{\eta_{i}(n)}} \right\rbrack.\mspace{20mu} {Let}}}}} & (4) \\{\mspace{79mu} {{{{\sum\limits_{i = 0}^{n - k}{{x\left( {n - i} \right)}{x\left( {n + 1 - i - l} \right)}}} = {\gamma_{i - 1}(n)}},\mspace{20mu} {{\sum\limits_{i = 0}^{n - k}\left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l - j} \right)}} \right\rbrack} = {\gamma_{j + 1 - 1}(n)}}}\mspace{20mu} {{{{for}\mspace{14mu} l} = 1},2,\ldots \mspace{14mu},{m;{j = 1}},2,\ldots \mspace{20mu},{{m.\mspace{20mu} {\sum\limits_{i = 0}^{n - k}{{x\left( {n - i} \right)}{\eta_{i}(n)}}}} \approx {0\mspace{14mu} \left( {{Data},{a\mspace{14mu} {noise}},{{being}\mspace{14mu} {uncorrelated}}} \right)}}}\mspace{20mu} {{{or}\mspace{20mu}\begin{bmatrix}{\gamma_{0}(n)} \\{\gamma_{1}(n)} \\\vdots \\{\gamma_{m - 1}(n)}\end{bmatrix}} = {{\begin{bmatrix}{\gamma_{1}(n)} & {\gamma_{2}(n)} & \; & {\gamma_{m}(n)} \\{\gamma_{2}(n)} & {\gamma_{3}(n)} & \; & {\gamma_{m + 1}(n)} \\\vdots & \vdots & \ldots & \vdots \\{\gamma_{m}(n)} & {\gamma_{m + k}(n)} & \; & {\gamma_{{2m} - 1}(n)}\end{bmatrix}\begin{bmatrix}{a_{1}(n)} \\{a_{2}(n)} \\\vdots \\{a_{m}(n)}\end{bmatrix}}.}}}} & (5)\end{matrix}$

Index k can be made variable for each n as long as k≧2m+1. In fact, kcan be increased as n increases.

Equation 5 can be rewritten in matrix-variable form:

γ(n)=Γ(n)a(n)   (6)

in which γ(n), Γ(n) and a(n) are defined in Eq. 5. Matrix Γ(n) is awell-known Hankel matrix and has many interesting properties includingrecursive inversion algorithms for large values of m.

Updated parameter vector computation and prediction Eq. 6 results in theoptimal evaluation of parameter vector a*_(m)(n) as follows:

a*(n)=(Γ(n))⁻¹ γ(n)   (7a)

or

a*(n)=(Γ^(T)(n)Γ(n))⁻¹ Γ^(T)(n)γ(n).   (7b)

Implementation of Eq. 7b in real time is very computation and storageintensive. A major contribution of the DMFA is a practical andcomputationally efficient solution of Eq. 7a and 7b.

Equations 7a and 7b yield the prediction algorithm:

x*(n+1)=x(n)a* ₁(n)+x(n−1)a* ₂(n)+ . . . +x(n−m)a* _(m)(n)   (8)

in which x*(n+1) is the predicted value of the future data value x(n+1),which is still not available. Accordingly,

γ₁(n+1)=γ_(l+1)(n)=x(n+1)x(n+2−l), l=1, 2, . . . , m.

The parameter a₀(n) is obtained as:

$\begin{matrix}{{a_{0}^{*}(n)} = {\left\lbrack \frac{\begin{matrix}{{\sum\limits_{i = 0}^{n - k}\left\lbrack {{x\left( {n - i} \right)}{d\left( {n + 1 - i} \right)}} \right\rbrack} -} \\{\sum\limits_{j = 1}^{m}\left( {\sum\limits_{i = 0}^{n - k}\left\lbrack {{x\left( {n - i} \right)}{d\left( {n + i - j} \right)}} \right\rbrack} \right)}\end{matrix}}{\sum\limits_{i = 0}^{n - k}{x\left( {n - i} \right)}} \right\rbrack.}} & (9)\end{matrix}$

The predicted data point at (n+1) is:

$\begin{matrix}{{{d^{*}\left( {\left( {n + 1} \right)/n} \right)} = {{a_{0}^{*}(n)} + {\sum\limits_{j = 1}^{m}\; {{d\left( {n + 1 - j} \right)}{a_{j}^{*}(n)}}}}}\left( {{one}\text{-}{step}\mspace{14mu} {prediction}} \right)} & (10)\end{matrix}$

The term “d*((n+1)/n)” represents the predicted value of d(n+1) giventhe data {d(i)}_(i=k) ^(n). An r-step prediction value can be computedas:

$\begin{matrix}{{{d^{*}\left( {\left( {n + r} \right)/n} \right)} = {{a_{0}^{*}(n)} + {\sum\limits_{j = 1}^{m}\; {{d^{*}\left( {n + r - j} \right)}{a_{j}^{*}(n)}}}}}{\left( {r\text{-}{step}\mspace{14mu} {prediction}} \right).}} & (11)\end{matrix}$

Index k is flexible and can be changed depending upon the process as itunfolds. In the case of a portfolio of more than one entity, Eq. 6 canbe rewritten as:

γ_(p)(n)=Γ_(p)(n)a _(p)(n).   (12)

Portfolio Optimization

The DMFA provides a major advance in optimal resource allocation, asdescribed herein. A portfolio consisting of different entities having astructure that is modeled by Eq. 1 and future values forecasted by Eq.11 can be optimized as follows:

Let

y_(d)(n+1)=Desired value of the portfolio at time (n+1);

p=the index for a respective of portfolio entity;

P=the number of portfolio entities;

d*_(p)((n+r)/n)=Forecasted values of entities, p=1, 2, . . . , P; r=1,2, . . . , R; and

w_(p)(n)=Weighting of the entitles in the portfolio, p=1, 2, . . . , P.

The value of the portfolio at instance n is:

$\begin{matrix}{{I(n)} = {\sum\limits_{p = 1}^{P}\; {{d_{p}(n)}{w_{p}(n)}}}} & (13)\end{matrix}$

The problem at hand is to compute optimal value of w_(p)(n+1) such thatthe portfolio value yields the desired value. Depending on the marketcondition with interest rate α, the desired portfolio value can bechosen as I_(d)((n+r)/n)=I(n)e^(arδt), in which δt is a time intervalbetween data samples, and α is the interest rate.

r is chosen to be greater than the number of entities p in the portfolioand the following portfolio optimization algorithm is arrived at:

$\begin{matrix}{\begin{bmatrix}{I_{d}\left( {\left( {n + 1} \right)/n} \right)} \\{I_{d}\left( {\left( {n + 2} \right)/n} \right)} \\\vdots \\{I_{d}\left( {\left( {n + R} \right)/n} \right)}\end{bmatrix} = {\quad{{\begin{bmatrix}{d_{1}^{*}\left( {\left( {n + 1} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + 1} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + 1} \right)/n} \right)} \\{d_{1}^{*}\left( {\left( {n + 2} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + 2} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + 2} \right)/n} \right)} \\\vdots & \vdots & \vdots & \vdots \\{d_{1}^{*}\left( {\left( {n + R} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + R} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + R} \right)/n} \right)}\end{bmatrix}\begin{bmatrix}{w_{1}\left( {n + 1} \right)} \\{w_{2}\left( {n + 2} \right)} \\\vdots \\{w_{p}\left( {n + R} \right)}\end{bmatrix}}\mspace{79mu} {or}}}} & \left( {14\; a} \right) \\{\mspace{79mu} {{I\left( {\left( {n + 1} \right)/n} \right)} = {\left\lbrack {D^{*}\left( {\left( {n + 1} \right)/n} \right)} \right\rbrack {w_{p}\left( {n + 1} \right)}}}} & \left( {14\; b} \right)\end{matrix}$

The optimal weights w_(p)(n+1) are computed as

w* _(p)(n+1)=[D* ^(T)((n+1)/n)D*((n+1)/n)]⁻¹ D* ^(T)((n+1)/n)I((n+1)/n).  (15)

FIG. 1 depicts one exemplary embodiment of a flow diagram of the subjectmatter disclosed herein.

Thus, the DMFA technique disclosed herein is a superior auto regression(AR) model as a general system of time-series realizations in order tocalculate the coefficients that fit the model for a better prediction.The system is solved via an inversion technique that avoids explicitinversion of more than a 2×2 matrix and computation ofhigher-dimensional determinants and co-factors. Furthermore, largenumber of parameters can be updated and the model re-fitted to reduceprediction errors. The technique can be further extended to solve for afinancial portfolio involving a plurality of securities. The minimummean-square algorithm used assures system stability by having poleswithin the unit circle. The matrix inversion implementation by thesubject matter disclosed herein is a significant advancement.

The minimum mean square algorithm has been used for predicting, updatingand optimizing a portfolio. Other optimization algorithms, such aslinear programming, non-linear programming and dynamic programming,could also be used to arrive at other algorithms for optimal forecast.

In an over-determined system, increasing n increases estimation errorsfor the same number of parameters m. Furthermore, increasing the numberof parameters m tends to confuse a system as information is inferredfrom a larger number of past states, thereby leading to inaccuratetracking for a system that is inherently a lower-order system thanassumed. This anomaly results in singular system matrices. An optimal m;n for a given input series may vary for a different input data timeseries. FIGS. 2-5 present comparisons between actual Standard and Poor's500 values and predicted values for different m and n. Accordingly,parallel computing is a powerful tool when the number of securities islarge. The optimal values for the quantities m and n may require somefine tuning for each particular situation.

In FIG. 2, curve 201 represents actual index data for the S&P 500 as ofJun. 2, 2006, and curve 202 represents predicted data provided by theDynamic Modeling and Forecasting Algorithm (DMFA) disclosed herein. Theabscissa of FIG. 2 represents time and the ordinate of FIG. 2 is the S&P500 index actual and predicted values. Prediction curve 202 utilizesthree (3) parameters and 20 data points.

In FIG. 3, curve 301 represents actual index data for the S&P 500 as ofJun. 2, 2006, and curve 302 represents predicted data provided by theDynamic Modeling and Forecasting Algorithm (DMFA) disclosed herein. Theabscissa of FIG. 3 represents time and the ordinate of FIG. 3 is the S&P500 index actual and predicted values. Prediction curve 302 utilizesthree (3) parameters and 30 data points.

In FIG. 4, curve 401 represents actual index data for the S&P 500 as ofJun. 2, 2006, and curve 402 represents predicted data provided by theDynamic Modeling and Forecasting Algorithm (DMFA) disclosed herein. Theabscissa of FIG. 4 represents times and the ordinate of FIG. 4 is theS&P 500 index actual and predicted values. Prediction curve 402 utilizesfour (4) parameters and 35 data points.

In FIG. 5, curve 501 represents actual index data for the S&P 500 as ofJun. 2, 2006, and curve 502 represents predicted data provided by theDynamic Modeling and Forecasting Algorithm (DMFA) disclosed herein. Theabscissa of FIG. 5 represents times and the ordinate of FIG. 5 is theS&P 500 index actual and predicted values. Prediction curve 502 utilizesfive (5)5 parameters and 40 data points.

FIG. 6 depicts an exemplary embodiment of an article of manufacture 600comprising a non-transitory computer-readable medium 601 having storedthereon instructions that, if executed, result in at least the subjectmatter disclosed herein. In one exemplary embodiment, computer-readablemedium 601 comprises a magnetic computer-readable medium. In anotherexemplary embodiment, computer-readable medium 601 comprises an opticalcomputer-readable medium. In yet other exemplary embodiments,computer-readable medium comprises a flash memory, a phase-changememory, and/or a chalcogenide-type memory or the like.

FIG. 7 depicts a functional block diagram of one exemplary embodiment ofan information-handling system 700 capable of determining multiplefrozen-parameter dynamic modeling and forecasting according to thesubject matter disclosed herein. Information-handling system 700 maytangibly embody any of several types of computing platforms.Additionally, information-handling system 700 may include more or fewerelements and/or different arrangements of elements than shown in FIG. 7,and the scope of the claimed subject matter is not limited in theserespects.

Information-handling system 700 may comprise one or more processors,such as processors 710 and/or processor 712, of which one or more maycomprise one or more processing cores. In one exemplary embodiment,processors 710 and 712 are coupled to one or more memories 716 and/or718 via a memory bridge 714, which may be disposed external toprocessors 710 and/or 712, or alternatively at least partially disposedwithin one or more of processors 710 and/or 712. Memory 716 and/ormemory 718 may comprise various types of semiconductor-based memory, forexample, a volatile-type memory and/or a nonvolatile-type memory. In oneexemplary embodiment, memory bridge 714 may couple to a graphics system720 to drive a display device (not shown) coupled toinformation-handling system 700.

Information-handling system 700 may further comprise an input/output(I/O) bridge 722 to couple to various types of I/O systems. I/O system724 may comprise, for example, a universal serial bus (USB)-type system,an IEEE 1394-type system, or the like, to couple one or more peripheraldevices to information-handling system 700. A Bus system 726 maycomprise one or more bus systems, such as a peripheral componentinterconnect (PCI) express-type bus or the like, to connect one or moreperipheral devices to information-handling system 700. A hard disk drive(HDD) controller system 728 may couple one or more hard disk drives, orthe like, to information-handling system 700, such as, a Serial ATA-typedrive or the like, or alternatively a semiconductor-based drivecomprising flash memory, phase-change memory, and/or chalcogenide-typememory or the like. A switch 730 may be utilized to couple one or moreswitched devices to I/O bridge 722, for example, Gigabit Ethernet-typedevices or the like.

Appendices

Appendix 1.

Given γ(n) and Γ(n), the vector a(n) is chosen to minimize

I=(Γ(n)a(n)−γ(n))^(T)(Γ(n)a(n)−γ(n)).

This minimization results in:

a*(n)=(Γ^(T)(n)Γ(n))⁻¹ Γ^(T)(n)γ(n)   (16)

Implementation of Eq. 16 in real time is very computation intensive. Incontrast, the DMFA technique disclosed herein provides a practical andcomputationally efficient solution.

Let

[Γ^(T)(n)Γ(n)]=B(n) (a symmetric matrix).

Γ^(T)(n)γ(n)=z(n)

yielding

a ^(T)(n)=B ⁻¹(n)z(n).

Appendix 2. Derivation of Recursive Algorithm for solution of B⁻¹(n).

Let

$\mspace{85mu} {{{B\text{?}(n)} = {\begin{bmatrix}{b_{11}(n)} & \ldots & {b\text{?}(n)} \\\; & \vdots & \; \\{b\text{?}(n)} & \ldots & {b\text{?}(n)}\end{bmatrix}\text{?}}},\mspace{79mu} {{B\text{?}(n)} = {\begin{bmatrix}{b_{11}(n)} & \ldots & {b\text{?}(n)} \\\; & \vdots & \; \\{b\text{?}(n)} & \ldots & {b\text{?}(n)}\end{bmatrix}{\text{?}.\text{?}}\text{indicates text missing or illegible when filed}}}}$

Define

$\begin{matrix}{{\left. \mspace{79mu} {{{{B\text{?}(n)} = \begin{bmatrix}{B\text{?}(n)} & {c\text{?}(n)} \\{c\text{?}(n)} & {b\text{?}(n)}\end{bmatrix}},{{c\text{?}(n)} = \begin{bmatrix}{b\text{?}(n)} \\\vdots \\{b\text{?}(n)}\end{bmatrix}}}\mspace{79mu} {{{B\text{?}(n)} = \begin{bmatrix}{A\text{?}(n)} & {e\text{?}(n)} \\{e\text{?}(n)} & {f\text{?}(n)}\end{bmatrix}},\mspace{79mu} {{B\text{?}(n)B\text{?}(n)} = {I\text{?}\mspace{14mu} {Identity}\mspace{14mu} {Matrix}}}}\mspace{79mu} {yielding}\mspace{79mu} \begin{matrix}{{{B\text{?}(n)A\text{?}(n)} + {c\text{?}(n)}} = {I\mspace{14mu} \left( {{Identity}\mspace{14mu} {matrix}} \right)}} \\{{{B\text{?}(n)e\text{?}(n)} + {c\text{?}(n)f\text{?}(n)}} = 0} \\{{{c\text{?}(n)e\text{?}(n)} + {b\text{?}(n)f\text{?}(n)}} = 1}\end{matrix}} \right\}.\text{?}}\text{indicates text missing or illegible when filed}} & (1.7)\end{matrix}$

Let

B _(l) ⁻¹ c _(l+1)(n)=g _(l+1)(n).   (18)

The solution of Eq. 17 is:

e _(l+1)(n)=−(f _(l+1)(n)g _(l+1)(n)

f _(l+1)(n)=(b _(l+1,l+1) −c _(l+1) ^(T)(n)g _(l+1)(n))⁻¹

A _(l+1)(n)=B _(l) ⁻¹(n)+f _(l+1)(n)(g _(l+1)(n)g _(l+1) ^(T)(n)).

Thus, B_(l+1) ⁻¹(n) is computed from B_(l) ⁻¹(n) and B_(l+1)(n). Whenl=m, the process is terminated. The implementation disclosed hereinrepresents a major advance for providing a practical and computationallyefficient solution.

Appendix 3.

Given:

B(n+1)=(B(n)+Δ(n))

Let

((B(n+0)⁻¹=(B(n)+Δ(n))⁻¹=(B(n)+εΔ(n))⁻¹, ε=1.

From Taylor series,

$\left( {{B(n)} + {{ɛ\Delta}(n)}} \right)^{- 1} = {{{T_{0}(n)} + {ɛ\; {T_{1}(n)}} + {ɛ^{2}{T_{2}(n)}} + \ldots} = {\sum\limits_{i = 0}^{\infty}\; {ɛ^{(i)}{T_{i}(n)}}}}$${{{or}\left( {\sum\limits_{i = 0}^{\infty}\; {ɛ^{(i)}{T_{i}(n)}}} \right)}\left( {{B(n)} + {{ɛ\Delta}(n)}} \right)} = {I.}$

Equating powers of ε,

T _(i)(n)=(−1)^(i) T _(i−1)(n)Δ(n)B ⁻¹(n), i=1, 2, . . .

T ₀(n)=B ⁻¹(n).

For

$\frac{{\Delta (n)}}{{B(n)}}1$B⁻¹(n + 1) ≈ B⁻¹(n) − B⁻¹(n)Δ(n)B⁻¹(n).

Although the foregoing disclosed subject matter has been described insome detail for purposes of clarity of understanding, it will beapparent that certain changes and modifications may be practiced thatare within the scope of the disclosed subject matter. Accordingly, theexemplary embodiments are to be considered as illustrative and notrestrictive, and the subject matter disclosed herein is not to belimited to the details given herein, but may be modified within thescope and equivalents of the present disclosure.

1. A system comprising: an input/output (I/O) device capable ofreceiving information relating to a data set; and a processing devicecoupled to the I/O device, the processor capable of determining apredicted future value of a data point by: defining a predicted datavalue for a future data value d(n+1) of a current data point d(n) asd*((n+1)/n), in which the current data point d(n) is part of the dataset, the data set comprises {d(i)}_(i=k) ^(n) and k comprises a minimumnumber of initial samples required before prediction can begin; settingd(n+i−l)=a ₀(n)+d(n−i−l)a ₁(n)+ . . . +d(n+1−i−l m)a _(m)(n)+ε_(i)(n) inwhich i=0, 1, 2, . . . , n−k, l=1, 2, . . . , m+1, n≧k+2m+1, k≧2m+1,coefficients {a_(i)(n)}_(l=1) ^(m+1) comprise a parameter process, andvariables {ε_(i)(n)}_(i=k) ^(n) comprise random noise having zero meanand being uncorrelated with the data d(n) in the data set; determiningx(n+1−i−l)=d(n+1−i−l)−d(n+1−i−l−1), andη_(i)(n)=ε_(i)(n+1)−ε_(i)(n); determining${x\left( {n + 1 - i - l} \right)} = {{\sum\limits_{j = 1}^{m}\; {{x\left( {n + 1 - i - l - j} \right)}{a_{j}(n)}}} + {\eta_{i}(n)}}$for l=1, 2, . . . , m, and i=k, k+1, . . . n; multiplying x(n+1−i−l) byx(n−i) and summing from i=0 to i=n−k to obtain${{\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l} \right)}} \right\rbrack} = {{\sum\limits_{j = 1}^{m}\; {\sum\limits_{i = 0}^{n - k}\; {\left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - {l\; j}} \right)}} \right\rbrack {a_{j}(n)}}}} + {\sum\limits_{i = 0}^{n = k}\; \left\lbrack {{x\left( {n - i} \right)}{\eta_{i}(n)}} \right\rbrack}}};$     setting$\mspace{79mu} {{{\gamma_{i - 1}(n)} = {\sum\limits_{i = 0}^{n - k}\; {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l} \right)}}}},{and}}$$\mspace{79mu} {{\gamma_{j + 1 - 1}(n)} = {\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l - j} \right)}} \right\rbrack}}$     for  l = 1, 2, …  , m, and  for  j = 1, 2, ⋯  , m;     setting$\mspace{79mu} {{{\sum\limits_{i = 0}^{n - k}\; {{x\left( {n - i} \right)}{\eta_{i}(n)}}} \approx 0};}$determiningx*(n+1)=x(n)a* ₁(n)+x(n−1)a* ₂(n)+ . . . +x(n−m)a* _(m)(n) in whichx*(n+1) is a predicted value of the future data value x(n+1); settingγ_(l)(n+1)=γ_(l+1)(n)=x(n+1)x(n+2−l) for l=1, 2, . . . , m; determininga predicted process parameter a*₀(n) as${{a_{0}^{*}(n)} = \left\lbrack \frac{{\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{d\left( {n + 1 - i} \right)}} \right\rbrack} - {\sum\limits_{j = 1}^{m}\; \left( {\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{d\left( {n + i - j} \right)}} \right\rbrack} \right)}}{\sum\limits_{i = 0}^{n - k}\; {x\left( {n - i} \right)}} \right\rbrack};$and determining a predicted data point at (n+1) as:${d^{*}\left( {\left( {n + 1} \right)/n} \right)} = {{a_{0}^{*}(n)} + {\sum\limits_{j = 1}^{m}\; {{d\left( {n + 1 - j} \right)}{{a_{j}^{*}(n)}.}}}}$2. The system according to claim 1, wherein the data set comprisesfinancial portfolio data of a portfolio, and the processor furthercapable of: setting a desired future value of the portfolio at time (n+1) to be y_(d) (n +1); setting a forecasted value of a data point of anentity in the portfolio to be d*_(p)((n+r)/n) for p=1, 2, . . . , P, andr=1, 2, . . . , R, in which R is selected to be greater than setting aweighting of each entity in the portfolio to be w_(p) (n) for p=1,2, . .. , P; setting a current value of the portfolio at instance n to be${{I(n)} = {\sum\limits_{p = 1}^{P}\; {{d_{p}(n)}{w_{p}(n)}}}};$determining a desired portfolio value to be I_(d)((n+r)/n)=I(n)e^(arδt)in which δt is a time interval between data samples, and α is aninterest rate; and optimizing the entities in the portfolio as$\begin{bmatrix}{I_{d}\left( {\left( {n + 1} \right)/n} \right)} \\{I_{d}\left( {\left( {n + 2} \right)/n} \right)} \\\vdots \\{I_{d}\left( {\left( {n + R} \right)/n} \right)}\end{bmatrix} = {\quad{\begin{bmatrix}{d_{1}^{*}\left( {\left( {n + 1} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + 1} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + 1} \right)/n} \right)} \\{d_{1}^{*}\left( {\left( {n + 2} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + 2} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + 2} \right)/n} \right)} \\\vdots & \vdots & \vdots & \vdots \\{d_{1}^{*}\left( {\left( {n + R} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + R} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + R} \right)/n} \right)}\end{bmatrix}\begin{bmatrix}{w_{1}\left( {n + 1} \right)} \\{w_{2}\left( {n + 2} \right)} \\\vdots \\{w_{p}\left( {n + R} \right)}\end{bmatrix}}}$ or asI((n+1)/n)=[D*((n+1)/n)]w _(p)(n+1); and determining an optimal weightw*_(p)(n+1) for each entity in the portfolio asw* _(p)(n+1)=[D*^(T)((n+1)/n)D*((n+1)/n)]⁻¹ D* ^(T)((n+1)/n)I((n+1)/n).3. The system according to claim 2, wherein the data set comprisesfinancial portfolio data.
 4. The system according to claim 1, whereinthe data set comprises financial portfolio data.
 5. The system accordingto claim 1, wherein the data set comprises seismic data.
 6. A method forpredicting a future value of a data point, the method comprising:defining a predicted data value for a future data value d(n+1) of acurrent data point d(n) as d*((n+1)/n), in which the current data pointd(n) is part of a data set {d(i)}_(i=k) ^(n) and k comprises a minimumnumber of initial samples required before prediction can begin; settingd(n+i−l)=a ₀(n)+d(n−i−l)a ₁(n)+ . . . +d(n+1−i−l m)a _(m)(n)+ε_(i)(n) inwhich i=0, 1, 2, . . . , n−k, l=1, 2, . . . , m+1, n≧k+2m+1, k≧2m+1,coefficients {a_(i)(n)}_(l=1) ^(m+1) comprise a parameter process, andvariables {ε_(i)(n))_(i=k) ^(n) comprise random noise having zero meanand being uncorrelated with the data d(n) in the data set; determiningx(n+1−i−l)=d(n+1−i−l)−d(n+1−i−l−1), andη_(i)(n)=ε_(i)(n+1)−ε_(i)(n); determining${x\left( {n + 1 - i - l} \right)} = {{\sum\limits_{j = 1}^{m}\; {{x\left( {n + 1 - i - l - j} \right)}{a_{j}(n)}}} + {\eta_{i}(n)}}$for l=1, 2, . . . , m, and i=k, k+1, n; multiplying x(n+1−i−l) by x(n−i)and summing from i=0 to i=n−k to obtain${{\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l} \right)}} \right\rbrack} = {{\sum\limits_{j = 1}^{m}\; {\sum\limits_{i = 0}^{n - k}\; {\left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - {l\; j}} \right)}} \right\rbrack {a_{j}(n)}}}} + {\sum\limits_{i = 0}^{n = k}\; \left\lbrack {{x\left( {n - i} \right)}{\eta_{i}(n)}} \right\rbrack}}};$setting${{\gamma_{i - 1}(n)} = {\sum\limits_{i = 0}^{n - k}\; {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l} \right)}}}},{and}$${\gamma_{j + 1 - 1}(n)} = {\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l - j} \right)}} \right\rbrack}$for l=1, 2, . . . , m, and for j=1, 2, . . . , m; setting${{\sum\limits_{i = 0}^{n - k}\; {{x\left( {n - i} \right)}{\eta_{i}(n)}}} \approx 0};$determiningx*(n+1)=x(n)a* ₁(n)+x(n−1)a* ₂(n)+ . . . +x(n−m)a* _(m)(n) in whichx*(n+1) is a predicted value of the future data value x(n+1); settingγ_(l)(n+1)=γ_(l+1)(n)=x(n+1)x(n+2−l) for l=1, 2, . . . , m; determininga predicted process parameter a*₀(n) as${{a_{0}^{*}(n)} = \left\lbrack \frac{{\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{d\left( {n + 1 - i} \right)}} \right\rbrack} - {\sum\limits_{j = 1}^{m}\; \left( {\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{d\left( {n + i - j} \right)}} \right\rbrack} \right)}}{\sum\limits_{i = 0}^{n - k}\; {x\left( {n - i} \right)}} \right\rbrack};$and determining a predicted data point at (n+1) as:${d^{*}\left( {\left( {n + 1} \right)/n} \right)} = {{a_{0}^{*}(n)} + {\sum\limits_{j = 1}^{m}\; {{d\left( {n + 1 - j} \right)}{{a_{j}^{*}(n)}.}}}}$7. The method according to claim 6, wherein the data set comprisesfinancial portfolio data of a portfolio, the method further comprising:setting a desired future value of the portfolio at time (n+1) to beγ_(d)(n+1); setting a forecasted value of a data point of an entity inthe portfolio to be d*_(p)((n+r) for p=1, 2, . . . , P, and r=1, 2, . .. , R, in which R is selected to be greater than setting a weighting ofeach entity in the portfolio to be w_(p)(n) for p=1, 2, . . . , P;setting a current value of the portfolio at instance n to be${{I(n)} = {\sum\limits_{p = 1}^{P}\; {{d_{p}(n)}{w_{p}(n)}}}};$determining a desired portfolio value to be I_(d)((n+r)/n)=I(n)e^(arδt)in which δt is a time interval between data samples, and α is aninterest rate; and optimizing the entities in the portfolio as$\begin{bmatrix}{I_{d}\left( {\left( {n + 1} \right)/n} \right)} \\{I_{d}\left( {\left( {n + 2} \right)/n} \right)} \\\vdots \\{I_{d}\left( {\left( {n + R} \right)/n} \right)}\end{bmatrix} = {\quad{\begin{bmatrix}{d_{1}^{*}\left( {\left( {n + 1} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + 1} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + 1} \right)/n} \right)} \\{d_{1}^{*}\left( {\left( {n + 2} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + 2} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + 2} \right)/n} \right)} \\\vdots & \vdots & \vdots & \vdots \\{d_{1}^{*}\left( {\left( {n + R} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + R} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + R} \right)/n} \right)}\end{bmatrix}\begin{bmatrix}{w_{1}\left( {n + 1} \right)} \\{w_{2}\left( {n + 2} \right)} \\\vdots \\{w_{p}\left( {n + R} \right)}\end{bmatrix}}}$ or asI((n+1)/n)=[D*((n+1)/n)]w _(p)(n+1) determining an optimal weightw*_(p)(n+1) for each entity in the portfolio asw* _(p)(n+1)=[D* ^(T)((n+1)/n)D*((n+1)/n)]⁻¹ D* ^(T)((n+1)/n)I((n−1)/n).8. The method according to claim 7, wherein the data set comprisesfinancial portfolio data.
 9. The method according to claim 6, whereinthe data set comprises financial portfolio data.
 10. The methodaccording to claim 6, wherein the data set comprises seismic data. 11.An article comprising: a non-transitory computer-readable medium havingstored thereon instructions that, if executed, result in at least thefollowing: defining a predicted data value for a future data valued(n+1) of a current data point d(n) as d*((n+1)/n), in which the currentdata point d(n) is part of a data set {d(i)}_(i=k) ^(n) and k comprisesa minimum number of initial samples required before prediction canbegin; settingd(n+i−l)=a ₀(n)+d(n−i−l)a ₁(n)+ . . . . +d(n+1−i−l m)a _(m)(n)+ε_(l)(n)in which i=0, 1, 2, . . . , n−k, l=1, 2, . . . , m+1, n≧k+2m+1, k≧2m+1,coefficients {a_(i)(n)}_(l=1) ^(m+1) comprise a parameter process, andvariables {ε_(i)(n)}_(l=k) ^(n) comprise random noise having zero meanand being uncorrelated with the data d(n) in the data set; determiningx(n+1−i−l)=d(n+1−i−l)−d(n+1−i−l−1), andη_(i)(n)=ε_(i)(n+1)−ε_(i)(n); determining${x\left( {n + 1 - i - l} \right)} = {{\sum\limits_{j = 1}^{m}\; {{x\left( {n + 1 - i - l - j} \right)}{a_{j}(n)}}} + {\eta_{i}(n)}}$for l=1,2, . . . , m, and i=k,k+1, . . . n; multiplying x(n+1−i−l) byx(n−i) and summing from i=0 to i=n−k to obtain${{\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l} \right)}} \right\rbrack} = {{\sum\limits_{j = 1}^{m}\; {\sum\limits_{i = 0}^{n - k}\; {\left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - {lj}} \right)}} \right\rbrack {a_{j}(n)}}}} + {\sum\limits_{i = 0}^{n = k}\; \left\lbrack {{x\left( {n - i} \right)}{\eta_{i}(n)}} \right\rbrack}}};$setting${{\gamma_{i - 1}(n)} = {\sum\limits_{i = 0}^{n - k}\; {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l} \right)}}}},{and}$${\gamma_{j + 1 - 1}(n)} = {\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{x\left( {n + 1 - i - l - j} \right)}} \right\rbrack}$for l=1, 2, . . . , m, and for j=1, 2, . . . , m; setting${{\sum\limits_{i = 0}^{n - k}\; {{x\left( {n - i} \right)}{\eta_{i}(n)}}} \approx 0};$determiningx*(n+1)=x(n)a* ₁(n)+x(n−1)a* ₂(n)+ . . . +x(n−m)a* _(m)(n) in whichx*(n+1) is a predicted value of the future data value x(n+1); settingγ_(l)(n+1)=γ_(l+1)(n)=x(n+1)x(n+2−l) for l=1, 2, . . . , m; determininga predicted process parameter a*₀(n) as${{a_{0}^{*}(n)} = \left\lbrack \frac{{\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{d\left( {n + 1 - i} \right)}} \right\rbrack} - {\sum\limits_{j = 1}^{m}\; \left( {\sum\limits_{i = 0}^{n - k}\; \left\lbrack {{x\left( {n - i} \right)}{d\left( {n + i - j} \right)}} \right\rbrack} \right)}}{\sum\limits_{i = 0}^{n - k}\; {x\left( {n - i} \right)}} \right\rbrack};$and determining a predicted data point at (n+1) as:${d^{*}\left( {\left( {n + 1} \right)/n} \right)} = {{a_{0}^{*}(n)} + {\sum\limits_{j = 1}^{m}\; {{d\left( {n + 1 - j} \right)}{{a_{j}^{*}(n)}.}}}}$12. The article according to claim 11, wherein the data set comprisesfinancial portfolio data of a portfolio, the method further comprising:setting a desired future value of the portfolio at time (n+1) to bey_(d)(n+1); setting a forecasted value of a data point of an entity inthe portfolio to be d*_(p)((n+r)/n) for p=1, 2, . . . , P, and r=1, 2, .. . , R, in which R is selected to be greater than setting a weightingof each entity in the portfolio to be w_(p)(n) for p=1, 2, . . . , P;setting a current value of the portfolio at instance n to be${{I(n)} = {\sum\limits_{p = 1}^{P}\; {{d_{p}(n)}{w_{p}(n)}}}};$determining a desired portfolio value to be I_(d)((n+r)/n)=I(n)e^(arδt)in which δt is a time interval between data samples, and α is aninterest rate; and optimizing the entities in the portfolio as$\begin{bmatrix}{I_{d}\left( {\left( {n + 1} \right)/n} \right)} \\{I_{d}\left( {\left( {n + 2} \right)/n} \right)} \\\vdots \\{I_{d}\left( {\left( {n + R} \right)/n} \right)}\end{bmatrix} = {\quad{\begin{bmatrix}{d_{1}^{*}\left( {\left( {n + 1} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + 1} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + 1} \right)/n} \right)} \\{d_{1}^{*}\left( {\left( {n + 2} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + 2} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + 2} \right)/n} \right)} \\\vdots & \vdots & \vdots & \vdots \\{d_{1}^{*}\left( {\left( {n + R} \right)/n} \right)} & {d_{2}^{*}\left( {\left( {n + R} \right)/n} \right)} & \ldots & {d_{p}^{*}\left( {\left( {n + R} \right)/n} \right)}\end{bmatrix}\begin{bmatrix}{w_{1}\left( {n + 1} \right)} \\{w_{2}\left( {n + 2} \right)} \\\vdots \\{w_{p}\left( {n + R} \right)}\end{bmatrix}}}$ or asI((n+1)/n)=[D*((n+1)/n)]w _(p)(n+1) determining an optimal weightw*_(p)(n+1) for each entity in the portfolio asw* _(p)(n+1)=[D* ^(T)((n+1)/n)D*((n+1)/n)]⁻¹ D* ^(T)((n+1)/n)I((n+1)/n).13. The article according to claim 12, wherein the data set comprisesfinancial portfolio data.
 14. The article according to claim 11, whereinthe data set comprises financial portfolio data.
 15. The articleaccording to claim 11, wherein the data set comprises seismic data.